6 REGULATED FUNCTIONS 145

a ,f h —-, a -f (A + 1) —- (Q^h^n - 2)

**""*""* i /*!"""™" i I

and have in each such interval a derivative equal to I or to 1; prove that the distance
of any two distinct elements of Mn { is > 2/(// 1). Similarly, consider the subset
Mfl of Mrt consisting of the 2W ' functions of M« which are equal to x a in the
interval [«,</ I (/> w)///K and for each function //1 Mi, consider the set of all
functions /< K such that //(,,v) (2/«) ^f(x)<g(x) for every .XT I. Use a similar
construction when (/> «)//• is not an integer.)

6. REGULATED FUNCTIONS

Let I be an interval in R, of origin a and extremity h (a or h or both
may be infinite), F a Banach space, We say that a mapping / of I into F
is a step-function if there is an increasing finite sequence (A^O^^W of points
of 1 (closure of I in K) such that .v(> •« a> \\ = h, and that /is constant in each
of the open intervals ],Y/ , x{ n [ (0 < / < n - 1 ).

For any mapping / of 1 into F* and any point A* € I distinct from A, we
say that /has a limit on the right if lim f(y) exists; we then write the

y* l,

y

limit /( .v +). Similarly we define for each point A" el distinct from a, the

limit on the left of /at A*, which we write /(A*—); we also say these limits
are one-sided limits of/. A mapping/of 1 into F is called a regulated function

if it has one-sided limits at every point of I, It is clear that any step-function
is regulated,

(7,6,1) In order that a mapping f of a compact Interval I «= [a, h] into ₯ he
regulated* a necexmry ami sufficient condition is that f he the limit of a uniformly
convergent sequence of stcp-functlonx,

(a) Necessity, For every integer n, and every .vel, there is an open
interval VOv) ^ ]>!*:), r(A*)[ containing A\ such that lljfa) ~/(r)|| < 1/w if
either both $, / are in ].v(.vX **[ n I or both in ].v, r(-v)[ n I. (''over I with a
finite number of intervals V(A*J), and let (^)(}^;»:m be the strictly increasing
sequence consisting of the points a* IK xt, y(xt) and z(.V|). As each cj is in some
V(.t|), cj f, | is either in the same V(.v/) or we have Cj M « z(A*f), for J < w - 1;
in other words If s^t are both in the same interval ]c^cjn[% then
II/CO **/(0|| < 1/w. Now define gn as the step-function equal to / at the
points Cj, and at the midpoint of each interval Jfy, fyfit* and constant in
each of these intervals. It is clear that ||/~#J < I//*.); show that if v (resp. H') is finite ut one point *» »it is